I discuss recent work on constructing gauge invariant quantum states in constrained Hamiltonian systems by averaging noninvariant states over the gauge group. Focussing on specific systems with a finite-dimensional phase space, I address convergence of the averaging for a noncompact gauge group, indefiniteness of the prospective physical inner product and the emergence of superselection sectors. These issues present technical challenges to developing group averaging towards background-independent theories such as general relativity.